2022 AMC 12B Problems/Problem 24
Contents
Problem
The figure below depicts a regular 7-gon inscribed in a unit circle.
What is the sum of the 4th powers of the lengths of all 21 of its edges and diagonals?
Solution (Complex numbers approach)
There are 7 segments whose lengths are , 7 segments whose lengths are
, 7 segments whose lengths are
.
Therefore, the sum of the 4th powers of these lengths is
where the fourth from the last equality follows from the property that
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution (Trig approach)
There are 7 segments whose lengths are , 7 segments whose lengths are
, 7 segments whose lengths are
.
Therefore, the sum of the 4th powers of these lengths is
where the second from the last equality follows from the property that
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~ ThePuzzlr
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)